May 16, 2016 - The time it takes for a particle with velocity v to cross a structure of radius R ...... with three separate programs which I have written in the Python.
Structural and dynamical properties of dark matter halos Gustav Collin Rasmussen DARK Cosmology Center Niels Bohr Institute May 16, 2016
Contents 1 Abbreviations
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2 Introduction
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3 Mergers in the universe 3.1 Phase mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Violent relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Double-power density models
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5 Generating initial structures 11 5.1 Eddingtons inversion method . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Osipkov-Merritt models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.3 Overview of structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6 Controlled numerical Simulations 6.0.1 I: G-perturbations . . . . . . . . . . . . . . . . . . . . . 6.0.2 IIa: Energy exchange (cartesian velocity perturbations) 6.0.3 IIb: Energy exchange (radial velocity perturbations) . . 6.0.4 IIc: Energy exchange (tangential velocity perturbations) 6.0.5 IId: Energy exchange (speed perturbations) . . . . . . .
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7 Statistical analysis and results 7.1 Time evolution . . . . . . . . 7.2 Selection of radial domains . . 7.3 Cuts and range of trust . . . 7.4 Unstable OM models . . . . .
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8 Conclusion
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A A note on simulation time
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B Side projects B.1 Cosmological SIMS . . . . . . . . B.2 Velocity distribution functions . . B.3 Line-of-sight overdensities . . . . B.4 The bumpy road to universalities
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C Functional analysis
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D Brief overview and description of analysis codes D.1 VDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Line-of-sight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 Removing gravitationally unbound particles from structures . . . . . . . . .
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E Background: Observational support to DM existence
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F The GADGET-2 code
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G Continuity Equation and Collisionless Boltzmann Equation
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H Derivation of Jeans Equation
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Proof that Tsallis q-fit converges to gaussian function
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J Datafiles for comparison K Analysis codes K.1 Read.py . . . . . . . . . . K.2 Sigma.py . . . . . . . . . . K.3 gamma_kappa_beta.py . K.4 VDF.py . . . . . . . . . . K.5 VDF_LOS.py . . . . . . . K.6 Remove_free_particles.py K.7 Energy_exchange.py . . .
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L The ΛCDM Model and DM candidates
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M Potential
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Abstract Whether or not the precise structure of each individual virialized DM halo in the universe today is sharply dictated by the laws of nature or a product of random mergers and collisions throughout the formation history of the universe is unknown [16]. In this sense it could be either random or determined by physics. Randomness is favored if all structures look different but physics is the fundamental driving of today’s structures if we can observe common trends and universalities. This is the fundamental question being addressed. This work investigates universalities of spherical dark matter structures in equilibrium by the study of collisionless N-body systems. These systems are simulated numerically in a controlled way using the N-body code GADGET-2. The equilibrium structures are generated using Eddingtons inversion method and Osipkov-Merritt models by the means of two Ccodes developed by Sparre (see [1] and [2]). Four side projects are carried out as well and can be seen in Appendix B. They address related questions, e.g. morphology of cosmological dark matter halos, how particle velocities are distributed throughout DM structures, comparison of simulated and observable quantities and finally looking for universal features in the rate of change of density profiles for different structures.
Acknowledgements This work could not have been possible without the great help of my thesis supervisors Steen H Hansen and Martin Sparre. Martin has been a solid support for programming issues along the way. I believe Steen must be one of the very few supervisors who will respond to technical questions over email even late in the evening. It has been very rewarding to work under Steen’s skilled supervision as the project took several interesting directions along the way. I am in much gratitude to the people at Dark Cosmology Center for advise and fruitful conversations. Thanks to my family and friends for all their support. This has been a wonderful appetizer for digging further into the realms of astrophysics. NB: The cover image is taken from the Illustris simulation and can be found at this adress: http://i1.ytimg.com/vi/vGpZU1LSsdY/0.jpg
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Abbreviations
BAO: Baryon Acoustic Oscillations BBN: Big Bang Nucleosynthesis CBE: Collisionless Boltzmann Equation CDM: Cold Dark Matter CE: Continuity Equation CGS: Centimeter-Gram-Second system CMBR: Cosmic Microwave Background Radiation DE: Dark Energy DF: Distribution Function DM: Dark Matter Edd: Eddingtons inversion method EOM: Equation Of Motion FWHM: Full Width Half Maximum (connected to Gaussian distribution) HBB: Hot Big Bang model HDM: Hot Dark Matter HE: Hydrostatic Equilibrium HQ: Hernquist density profile JE: Jeans Equation LHS: Left Hand Side ln: Natural logarithm (Logarithm with base e, Euler´s number) log: Logarithm with base 10 MKS: Meter-Kilogram-Second system (SI-units) MOND: MOdified Newtonian Dynamics MW: Milky Way galaxy NFW: Navarro-Frenck-White density profile N-D: N-dimensional/N-dimensions (N Z+ ) OM: Opsikov-Merritt model pct: Percentage pdf: Probability distribution function PDF: Probability Density Function PPSD: Pseudo Phase Space Density RHS: Right Hand Side ROI: Radial Orbit Instability SIMS: Simulations SM: Standard Model (of particle physics) SMC: Cosmological Standard Model STD: STandard Deviation (of normal-distribution) SUSY: SUper SYmmetry VDF: Velocity Distribution Function WDM: Warm Dark Matter wrt: with respect to ΛCDM: Cosmological Standard Model (The so-called concordance cosmology with DE and CDM ) Z+ : Positive integer numbers
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2
Introduction
The purpose of this section is to elaborate a bit on the abstract and to provide a clear picture of the structure for this thesis. Starting with speculations on the nature of dark matter based on current observational constraints we can set up the appropriate theoretical framework and introduce some important concepts to describe collisionless systems. This chapter ends with the general notion of an attractor and some ideas for where to search for it in terms of dark matter systems (if it exists).
Advancements has been made in narrowing down the window of theoretically allowed DM particles through modern observational constraints (upper bounds to DM cross sections are lowered further as more and more DM structures aids in the vast collection of constraining structures, e.g. the bullet cluster where gravitational lensing of background objects shows two huge parts of hidden mass as remnants of the merger between two galaxy clusters with their individual dark halos (the hot X-ray gas remains in the center as it is subjected to numerous collisions intrinsically). Most astronomers believe dark matter candidates to be in the form of one or several new types of elementary particles (existing outside the current standard model of particle physics). Theoretical framework Quantum effects are negligible Utilizing Ehrenfests theorem since the mean inter-particle distance in each simulation in the present work is larger than the particles Debroglie wavelength λ = hp , we can neglect any quantum mechanical treatment and rely solely on a classical mechanical treatment for our purpose here. It is not known if particle collisions are important in real halos. Probably this will not play a role in Milky way-like galaxies but in dwarf galaxies it might [12]. The Newtonian limit is applicable General Relativity (GR) can also be safely neglected here, as all three conditions for taking the Newtonian limit is met (v ra I am using a C code made publicly available by Martin Sparre [2].
Figure 6: β for IC of structure CS_6 shown together with the analytical expression β(r) = r2 . It is seen that the analytical expression agrees well with the computed velocity ra2 +r2 anisotropy for the anisotropy radius ra = 4.0, which is the value specified for the IC of the CS_6 structure. 20 radial bins is used. Analytical expression of β: The β expression for an OM model is simply given by β(r) = set ra = 1, the analytical form becomes just β(r) = agree very well with the IC of OM model CS_6.
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r2 1+r2
r2 ra2 +r2
so if we choose to
which can be seen in figure 6 to
Overview of structures
All structures can be divided into two categories, namely stable or unstable (see tables). Most of them follow Hernquist density profiles (see previous section) but three follow the (α, β) = (0, 5) density profile. The unstable structures are dealt with in the later section ’Unstable OM models’. ICs have been chosen so that they together span a large volume in the 3D Jeans parameter solution space (β, γ, κ). This creates an opportunity to test if they all keep spanning a large volume or flow towards an attractor. Since all HQ structures have a total mass of unity but vary in particle number, the more numerous structures have particles with lower mass. Variations in anisotropy radius 17
are carried out while setting up the different OM models. Gravitational softening is varied so that it decreases by a factor when particle number is increased (see tables). For all 1 simulations, the scale radius rs = 1 is used and ρ0 = 2π (which ensures that all HQ structure has a total numerical mass value of 1). The type of simulation each structure is undergoing is indicated by I: ’Instantaneous change in the gravitational potential’ and II: ’Energy exchange’ (See later sections for description of I and II). N IC ρ ra SIMS ∆G
A 106 Edd HQ 0.1 I 20%
B 106 Edd HQ 0.1,0.02 I,II 5%
CS1 104 OM HQ 1.33 0.1 II -
CS2 104 OM HQ 1.5 0.1 -
CS3 104 OM HQ 2.0 0.1 -
CS4 105 OM HQ 1.5 0.04 I,II 20%
CS5 105 OM HQ 2.0 0.04 I,II 20%
CS6 105 OM HQ 4.0 0.04 I, II 20%
DS1 105 OM (0,5) 2.0 0.04 I,II 20%
D2 105 Edd (0,5) 0.1,0.04 I,II 20%
E 106 OM HQ 2.0 0.02 I,II 20%
Table 1: Stable structures. From top to bottom this table states the name of each stable structure, the total number of particles this structure holds (N), the method utilized for setting up the initial conditions (IC’s), the type of double-power law model used for creating this structure, for Osipkov-Merritt (OM) models the anisotropy radius is then given. Next follows the value of the gravitational softening, , the types of simulations the different structures are subjected to (I: G-perturbations or II: Energy exchange), for sim. I the percentage of variation to Newtons gravitational constant is then stated and finally for sim. II the maximum percentage (most particles are subjected to a variation in velocity of size smaller than this percentage since random uniform numbers in the given range dictate the size of each velocity kick, see the later section ’II: Energy exchange’ for more details about this sim.) of variation to each particles velocity is stated.
N IC ρ ra SIMS ∆G
C1 104 OM HQ 1.0 0.1 II -
C2 104 OM HQ 0.2 0.1 -
C3 104 OM HQ 0.1 0.1 -
C4 105 OM HQ 1.0 0.1,0.4 I,II 20%
C5 105 OM HQ 0.2 0.1,0.4 I,II 20%
C6 105 OM HQ 0.1 0.1,0.4 I,II 20%
D1 105 OM (0,5) 1.0 0.1,0.4 I,II 20%
Table 2: Unstable structures. From top to bottom this table states the name of each stable structure, the total number of particles this structure holds, the method utilized for setting up the initial conditions (IC’s), the type of double-power law model used for creating this structure, the anisotropy radius, the value of the gravitational softening, , the types of simulations the different structures are subjected to (I: G-perturbations or II: Energy exchange), for sim. I the percentage of variation to Newtons gravitational constant is then stated and finally for sim. II the maximum percentage (most particles are subjected to a variation in velocity of size smaller than this percentage since random uniform numbers in the given range dictate the size of each velocity kick, see the later section ’II: Energy exchange’ for more details about this sim.) of variation to each particles velocity is stated. These structures are all OM models and they are unstable because their anisotropy radii are too small. In order to have stability the OM structures need to meet the condition ra > 1.33 which neither of these structures fulfill.
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Centralization of halos All structures are centralized both wrt each particles cartesian coordinates and wrt its cartesian velocities. The center is first found by getting the position of the gravitational potential minimum and the central coordinates and velocities is denoted as xc , yc , zc , vxc , vyc and vzc respectively.
Figure 7: All three cartesian coordinates are centralized around the halo center by subtraction of each individual particles position by the central coordinates xc , yc and zc . Our Initial conditions are set up and ready to run. The next section introduces two principal ways to mimic the galaxy merger related effects. We will see how violent relaxation and phase mixing can be emulated by both sim. I and sim. II.
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Controlled numerical Simulations
Two different types of simulations are involved in this work, namely cosmological (see side projects in Appendix B) and controlled sims (described in this section). Both types are produced by using the GADGET-2 code (Springel 2005). The cosmological simulation is just analyzed, whereas the controlled simulations are both performed (inside a non-cosmological Newtonian box) and subsequently analyzed. By the name controlled it is meant that the sim. is more local and does not take an expanding spacetime into account. This enables a more detailed study where hierarchical clustering, mergers, build-up of 19
halo sub-structure and accretion effects does not wash out the potential attractors of interest. The controlled sims is comprised of two types, namely I (An instantaneous change of the gravitational potential) and II (Energy exchange) which is further varied in four principal ways (cartesian, radial, tangential and total velocity perturbations).
When looking for universalities it is crucial to start from many different IC’s. It is equally important to perturb these IC’s in a variety of ways in order to make sure any found universalities is not an artifact of some artificial computer method. For this reason structures are subjected both to Sims. of type I and II. Both types have an underlying structure in common: Structures are perturbed and then gravity is allowed to work on them during subsequent simulation. This perturbation→simulation method is kept under all numerical experiments in this work. They both serve to mimic the effects dominant during galaxy mergers since these dramatic events can alter characteristic features of the galaxies surrounding dark matter halos. It will then become more clear if these halos are driven towards any preferred state or configuration.
Particle number Particle number is steadily increased from 104 to 105 and finally all the way to 106 . To compute for example the velocity dispersion σ or the density ρ for a structure divided into 20 radial bins, and to get a good statistical result, 5 · 102 particles is needed in each bin and therefore a minimum of 104 particles must be included. However, in order to determine derived or constructed quantities such as κ or γ, a minimum of 5 · 103 particles in each of the 20 bins is more favorable and hence a total of 105 particles is wanted. Looking at 2.nd order derivatives as for example in the project ’Bumpy road to universalities’, even more particles must be included. A minimum of 5 · 104 particles in each of the 20 bins is more favorable and hence a total of 106 particles is wanted. Gravitational softening For most cosmological simulations the softening depends upon the particle number in the following way [10]: 1 3 ≥( )3 d (40) 2 800πN 1 d. For the controlled sims. it where d is the mean particle distance. For N = 5, ≥ 30 1 is simply assumed that scales as N 3 . In Sparre and Hansens paper (http://arxiv. org/abs/1210.2392) a slightly smaller softening of 0.0050 were used in both the Gperturbations and the energy exchange simulations, It is therefore interesting to see if the attractor also exists if the above softening-values are used. This is expected to be the case, as a slightly larger softening only affects the very inner part. As a rule of thumb the softening must become 10 times larger when the particle number increases a thousand times [11].
Range of Jeans parameters For an Edd structure, β = 0 initially, and for an OM structure, β ∈ [0, 1]. So the range of β is set to [0, 1]. For the density profile ρHQ , γ ∈ [−3, −1], and for the density profile ρ0,5 , γ ∈ [−5, 0]. The γ-range thus depends on the density profile under consideration.
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6.0.1
I: G-perturbations
To mimic the proces of violent relaxation during galaxy and cluster mergers this type of sim. creates an instantaneous change in the gravitational potential. Such a change perturbs the acceleration of each particle. This section gives a technical description of exactly how each structure was perturbed wrt the gravitational potential during this type of sim. The section gives an overview of a specific pattern of G-perturbations and following simulation, which is repeated a certain number of times, as well as an overview of the simulation times used during each step.
The purpose of the simulations of type I is to resemble the proces of violent relaxation dΦ during mergers. But instead of using the relation describing violent relaxation dE dt = dt , where a time-dependence of the gravitational potential (Φ) exists, a time variation of Newtons gravitational constant is applied, G(t). This effectively produces the same effect. After each perturbation of G from its standard value, the structures are simulated. This pattern continues for a certain amount of time until a new stable configuration is found. Practically, changes are made to the value of Newtons gravitational constant G from its 2 standard value of G = 6.67 · 10−11 Nkgm2 here corresponding to G = 1 in non-SI units, to either 1.2 or 0.8 times the standard value, corresponding to an increase or decrease by 20 %, or 1.05 to 0.95 times the standard value, corresponding to an increase or decrease by 5 %. Either 20 or 5 pct is added or subtracted systematically from G in a certain pattern (see table 3). The Simulations always start out with G = 1 in Run 0 which last for 100 simulation times. This is done to give the structure time to work under its own gravity and make sure the IC’s are set up correctly. It provides a way to see that the IC is in a stable configuration. The pattern from Run 1 to Run 5 is then repeated a certain number of times (see table 4). Ending up with runs of G = 1 for longer time gives the structures better chance to reach stable equilibria (see tables 5-7). The idea behind changing the G-value by a smaller amount (5 pct.) from the standard value (as is the case with sim B) is to have the structure depart less from equilibrium during the simulations. It is seen later on in the analysis of the velocity distribution functions that these are not so different if the perturbations are larger or smaller, but the γ vs. β and γ + κ vs. β plots look slightly different if G is perturbed by 20% or by 5%. GADGET-2 makes a certain number of runs with varying values assigned to the gravitational constant G (Corresponding to perturbing the gravitational potential, i.e. An instantaneous change in Φ ). Most of these runs each produce 6 snapshot files, except for the last run in each simulation, where the time is increased from Timemax = 100 to Timemax = 2300 (giving 93 snapshots). The method of varying the potential corresponds to changing the accelerations of the individual particles. It is then interesting to see how the structures will respond, more on this in the next section: ’The attractor’. When the gravitational constant is lowered to the value of G = 0.8 and a structure is simulated by Gadget-2 for 100 time units, an expansion of that structure naturally occurs. When the gravitational constant is raised to the value of G = 1.2 and the structure again is simulated for 100 time units, A contraction of this structure now happens. This type of simulation where the structure pulsates is reminiscent of violent relaxation.
Particle tracking In order to test whether or not structures subjected to Sim. I has the tendency to expand systematically in the central region and throw out particles when pulsating as the effect of changing the gravitational potential, a few test particles has been chosen and 21
Simulation Run No. 0 1 2 3 4 5
A, C4 -C6 , CS4 -CS6 , D1 , D2 and DS1 G-value 1.0 1.2 0.8 1.0 1.0 1.0
B G-value 1.0 1.05 0.95 1.0 1.0 1.0
Table 3: Simulation I pattern. First this table states the name of each structure. The B structure is placed alone as it has a unique percentage of change to G of only 5%, whereas the others all have G-perturbations to the order of 20%. For each run a simulation time of 100 is used (corresponding to one dynamical time at r = 12.7rs ). Left column numerates the different simulation runs. For all structures undergoing this type of simulation they are initially set up in the GADGET-2 code to be subjected to gravity with a potential corresponding to the standard value of G (here it means G=1). After the zeroth run the simulation is paused and an increase of the G value (corresponding to a stronger gravitational potential) is applied, after which the simulation is continued. After the next 100 simulation times the G value is decreased (corresponding to a weaker potential) and the simulation is allowed to continue yet again for the next 100 simulation times. Finally G remains at the standard value for 3 · 100 simulation times. This whole pattern or ’chunk’ from run 1 to run 5 is then repeated a different number of times which can be seen in the following table.
Simulation No. repetitions of pattern
A 9
B and E 39
C4 -C6 and CS4 -CS6 9
D1 , D2 and DS1 9
Table 4: Pattern repetitions. The ’chunks’ of G-perturbations and subsequent simulations are repeated for the above stated number of times for all the different structures in question.
Run 46 47 48
Simulation A, C4 -C6 and CS4 -CS6 G 1.2 0.8 1.0
TimeMax 100 100 2300
Table 5: Final runs. After the series of pattern repetitions shown before, for the here mentioned structures three final perturbation → simulation experiments are run: G is increased by 20% and structures are subjected to their self-gravity for 100 sim. times, G is decreased by 20% and structures are run for 100 sim. times and finally G is kept at unity and structures are run for 2300 sim. times in order to give the outer part of the structures more time to reach stable equilibria.
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Run 196 197 198 199
Simulation B and E G 1.05 0.95 1.0 1.0
TimeMax 100 100 2300 2300
Table 6: Final runs. After the series of pattern repetitions shown before, for the here mentioned structures three final perturbation → simulation experiments are run: G is increased by 5% and structures are subjected to their self-gravity for 100 sim. times, G is decreased by 5% and structures are run for 100 sim. times and finally G is kept at unity and structures are run for 2 · 2300 sim. times Run 49
Simulation D1 , D2 and DS1 G 1.0
TimeMax 2300
Table 7: Final run. After the series of pattern repetitions shown before, for the here mentioned structures one final perturbation → simulation experiments is run: G is kept at unity and the structures are run for 2300 sim. times tracked through a few different time-slices/snapshot-files of the simulations (See figure 10). In conclusion, there is no major systematic tendency of outward drift but only a small effect of turning initial cuspy profiles into profiles with small cores. Total velocity profile As all structures have a scale-radius of unity, rs = 1, the total velocity distribution is centered around unity as well (See figure 11). Over time however, substructure is created as we notice the emergence of filaments or ’fingers’, each finger corresponding to an increase in the gravitational potential thus increasing individual particle accelerations and their contribution to the overall total velocity distribution is to drift outwards towards larger radii in this characteristic filament pattern. (see figure 12 and 13). Final density Figure 14 shows the flattening of structures as their density profiles go from cuspy to cored over the duration of the simulations. Final structures in average have cores with a r r size of log r−2 = −0.5, i.e. r−2 ≈ 0.32 or put in other words the cores span 32 % of r−2 , the radius where γ = −2 (for a HQ profile this equals the scale radius, r−2 = rs ). Structures become anisotropic over time In figure 15 the effect this sim. type I has on velocity anisotropy can be seen. All structures will over time grow radially biased wrt. the β-profile no matter if they belong to Edd or OM ICs. Density slope extremas Figure 16 reveals an interesting feature of final γ shapes. notice in the outer part how there seem to be distinct bumps and wiggles. This is explored further in the side project found in App. B.4 ’The bumpy road to universalities’ where different normalizations are carried out to look for universalities in the final γ-profiles which seem to poses up to 6 unique local extremas (3 minima and 3 maxima). Evolution of radial velocity dispersion Figure 17 compares initial and final κ-profiles. Particularly the two larger structures A and B both with N = 106 particles have spread out into a larger area. This indicates an increase of the radial velocity dispersion which is the result of the pulsating that structures undergo during these acceleration perturbations. that effect is much more visible in a large 23
Figure 8: Time evolution of radius for a few chosen particles (ID’s 105 , 2 · 105 , 3 · 105 and 4 · 105 ) is plotted for selected snapshot files (IC, 5_005, 10_005, 40_005 and 48_009) of Sim. I for structure A. The figure shows that there is no systematic tendency of neither outward nor inward drift as indicated by all four particles changing radius in a stochastic manner as time evolves. This is expected due to the pulsating effect the G-perturbations has on structures.
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structure holding more particles, as these have better statistics to give a more complete look of the radial velocity spread.
r r Figure 9: Plot of vtot vs. r−2 and log( r−2 ) respectively for IC of A. Analyzed out to a radius of Rlim = 104 . Notice how the majority of particles are clustered around the value r log( r−2 ) = 0.
6.0.2
IIa: Energy exchange (cartesian velocity perturbations)
Using structures holding a variety of initial conditions, this simulation type mimics effects found in galaxy mergers. Specifically, for all of these structures (named as B, CS4 , CS5 , CS6 , DS1 , D2 and E) the total velocities inside spherical bins is perturbed for each of its particles by multiplying each of those given particles cartesian velocities (vx , vy and vz ) by random numbers in different ranges, firstly the range [0.8, 1.2] (corresponding to perturbing the kinetic energies by random numbers in the range 12 · [0.64, 1.44]), taken from the continuous uniform distribution. This is a symmetric probability distribution where all intervals of the same length between 0.8 and 1.2 are equally probable. Following this exchange of kinetic energies between particles in fixed spherical bins, a normalization factor is applied to the total velocities, which ensures energy conservation. After such a velocity ’kick’ the structures are simulated under their self gravity allowing them to flow a bit under their new velocities. This procedure is then repeated for a total of 40 runs (with one initial run where there is no perturbation. This allows the OM models to reach equilibrium, as 25
r r Figure 10: vtot vs. r−2 and log( r−2 ) respectively for sim. I of structure A, after 1100 simulation timesteps (corresponding to 11 · tdyn at r = 13.7 · rs ). Analyzed out to a radius of Rlim = 104 . A very slight shift of the main part of the structure towards larger radii is seen. Two velocity ’fingers’ or velocity filaments can now be seen at larger radii. They are a direct response to the perturbation where Φ is first increased and then decreased due to a change in Newtons gravitational constant. This greatly increase particles velocities at large radii, forcing some of them to become gravitationally unbound. Two segments of ’G-chunks’ have thus been run up to this stage as can be seen from simply counting the velocity filaments
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r r Figure 11: vtot vs. r−2 and log( r−2 ) respectively for sim. I of structure A, after 4970 simulation timesteps (corresponding to 49.7 · tdyn at r = 13.7 · rs ). Analyzed out to a radius of Rlim = 104 . The shifting of the main part of the structure towards larger radii remains present but is still not very significant. A much larger time span between this figure and the previous of ∆tsim = 4970 − 1100 = 3870 (or 38.7 · tdyn at r = 13.7 · rs ) means that far more ’G-chunks’ have been run at this stage, in fact ten of them, and again counting the velocity filaments we see this to be illustrated by the existence of ten filaments.
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r Figure 12: ρ vs. log( r−2 ). The IC profiles are all cuspy (apart from the DS1 and Soft_D2 structures which have unique density profiles with inner and outer slope zero and minus 5 respectively); i.e. their inner density profiles are very steep. As time evolves all profiles develop small cores in the very inner region. This can be easily understood through some simple dynamical arguments; Consider the few central particles that just happens to have a very high velocity greater than the escape velocity of the halo. Increasing the gravitational potential and then letting the simulation run will make these central particles travel outwards as the now stronger potential will tend to on average pull the particles into, and through the center (as they do not experience any collisions). Once through the center their travel continues out to large radii from there they will move with constant speed away from the structure forever. Hence the catchphrase ’Once out, always out’ applies to these high velocity central particles. This can be thought of as the main reason for the flattening of initially cuspy or steep density profiles at small radii. Structures with N = 106 particles (A, B and E) are divided into 50 bins which are logarithmic in radius, whereas structures with N = 105 particles are divided into just 20 bins to ensure a sufficient number of particles in each bin. The structures are cut off at a radius of Rlim = 32 · rs corresponding to the log10 value of 1.5
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r ). By comparing IC with final products wrt the velocity anisotropy Figure 13: β vs. log( r−2 parameter β it is seen that structures which are initially isotropic (A, B and Soft_D2, created by Eddingtons inversion method) become radially biased over time and the structures with initial velocity anisotropy (CS4, CS5, CS6, DS1 and E, which are OM models) remain anisotropic in the outer part as time goes. The inner part of the final structures are dominated by large fluctuations due to Poisson noise and it is therefore not possible to say anything about this region for certain apart from the fact that the fluctuations are centered around β = 0 so the structures could be ergodic in the inner part. The span in the β parameter for IC and final products are approximately [−0.2; 1.0], which can also be seen later in the (β,γ)- and (β,γ + κ)- plots. Structures with N = 106 particles (A, B and E) are divided into 50 bins which are logarithmic in radius, whereas structures with N = 105 particles are divided into just 20 bins to ensure a sufficient number of particles in each bin. The structures are cut off at a radius of Rlim = 32 · rs corresponding to the log10 value of 1.5
29
r Figure 14: γ vs. log( r−2 ). Structures with N = 106 particles (A, B and E) are divided into 50 bins which are logarithmic in radius, whereas structures with N = 105 particles are divided into just 20 bins to ensure a sufficient number of particles in each bin. The structures are cut off at a radius of Rlim = 32 · rs corresponding to the log10 value of 1.5
30
r Figure 15: κ vs. log( r−2 ). Structures with N = 106 particles (A, B and E) are divided into 50 bins which are logarithmic in radius, whereas structures with N = 105 particles are divided into just 20 bins to ensure a sufficient number of particles in each bin. The structures are cut off at a radius of Rlim = 32 · rs corresponding to the log10 value of 1.5
31
Figure 16: γ vs. β. Left panel shows the initial conditions and right panel the final products after sim. I of structures A, B, CS4 , CS5 , CS6 , DS1 , D2 and E. Notice how the IC spans a large volume of the Jeans parameter space whereas the final products are attracted to a 1-dimensional s-shaped attractor curve which effectively reproduces results found by others [4]. This attractor can be described analytically as γ + κ = −8β for the inner part and γ + κ = −0.7 − 4β for the outer region. For reference the final structures are plotted together with the linear relation found by Hansen and Moore (solid black line, β = −0.2(γ + 0.8)) as well as data from [4] which are in very good agreement.
32
Figure 17: γ + κ vs. β. Left panel shows the initial conditions and right panel the final products after sim. I of structures A, B, CS4 , CS5 , CS6 , DS1 , D2 and E. Notice how the IC spans a large volume of the Jeans parameter space whereas the final products are attracted to a 1-dimensional s-shaped attractor curve which effectively reproduces results found by others [4]. This attractor can be described analytically as γ + κ = −8β for the inner part and γ + κ = −0.7 − 4β for the outer region. For reference the final structures are plotted together with the linear relation found by Hansen and Moore (solid black line, β = −0.2(γ + 0.8)) as well as data from [4] which are in very good agreement.
33
they are not perfectly equilibrated when set up as OM models. It is done for Edd structures as well to be certain of equilibrium before any perturbations take effect). After the first 10 runs, the random number range is changed from [0.8, 1.2] to [0.95, 1.05] to look for more subtle effects in run 11 → 20. For run 21 → 40 the random range [0.7, 1.3] is applied, and the runs 31 → 40 are given longer timespans to allow structures more flow (See table). This way of exchanging particle energies is reminiscent of phase mixing.
Figure 18: This simple cartoon illustrate the effects of the energy exchange simulation on the total velocity distribution. I:qThe initial distribution of the total velocity given in terms of cartesian coordinates, v = vx2 + vy2 + vz2 . It follows a Gaussian shape with mean velocity equal to zero. II: Velocity kicks are now given to each particle in the structure, by multiplication of the initial velocities with a uniformly distributed random number in the range [0.8, 1.2]. This has the effect of shifting the mean of the distribution towards higher values. Furthermore, some of the particles with very high velocities (and further away from the structures center, where the potential is weaker) now become gravitationally unbound. This is illustrated by the red bump at the right tail of the right-shifted, randomized velocity curve shown in purple. The total energy is now higher than what it was initially. As the velocities were perturbed by factors in the range [0.8, 1.2], the kinetic energies K ∝ v 2 will have their initial values changed by factors in the range [0.64, 1.44] giving a new kinetic energy span of [1-0.36, 1+1.44] times the initial kinetic energies. III: In order to conserve energy, a normalization factor is multiplied with the initial energy, and used to normalize particle velocities after the previous randomization, so that the velocity mean of zero is regained. A small bump at the right tail remains however, which is seen on the final curve shown in pale blue. This whole process of velocity-kick and energy normalization is followed by a phase of flow after which the pattern is repeated (kick → flow). A parallel control simulation (with 20 runs) is performed where the energy is not exchanged in this manner. No alterations are done; i.e. G remains at unity throughout and no velocity kicks are given by hand to any particle. The purpose of this control simulation is to make sure the structures stay in equilibrium and provide a means of comparison for the perturbed structures. Binning of structures is done in the following way: First the radius of all particles to the halo center is sorted from smallest to largest value and this sorting is then applied to the positions and velocities as well as the potential. Now number bins are created so that each bin holds exactly 500 particles and runs from smallest to largest radii. The volume of each bin is hence generally different and depends on the mean inter-particle distance which grows as radius increases. It is therefore the case that outermost bins take up more space than innermost number-bins. Inside each number bin the perturbation and normalizing 34
Run No. 0 1-10 11-20 21-30 31-40
velocity kicks 0% 20 % 5% 30 % 30 %
Flow time 100 100 100 100 500
Table 8: Simulation II pattern. The left column shows the number of run in question. The middle column gives the maximum percentage of velocity kick to each particle along each of the three cartesian velocity directions, vx , vy and vz . Finally the right column states the time allowed for structures to flow freely under their self gravity. Run 0 takes place without any prior perturbation and lasts for 100 simulation times. In short, all structures are given medium velocity kicks before the first 10 runs following perturbations (run 1 → 10), while they are given small kicks before run 11 → 20, large kicks are given for run 21 → 40. Each of the runs from 1 → 30 lasts for 100 simulation times, but the last 10 runs (31 → 40) are given 500 simulation times each. of particle velocities and energies can then be performed. Finally saving the perturbed structures into a new snapshotfile, this is then ready to be read and simulated by the GADGET-2 code. More specifically about the perturbations: Starting out by giving each particle a velocity kick by a random number in the above stated range for each of its three cartesian velocity directions provideqthe particles with a new velocity given by 2 2 2 vrand = vx,rand + vy,rand + vz,rand where for example vz,rand = vzinitial + ∆vz = vzinitial 1 + rand[0.8, 1.2] and rand[0.8, 1.2] is a uniform random number between 0.8 and 1.2, thus perturbing each of the cartesian velocity components by maximally 20 % that the √which in turn means √ maximum perturbation to the total velocity vector can be 1200% = 20 · 3% ≈ 34.6%. The kinetic energy becomes 2 Krand = 12 vrand In order to ensure energy conservation normalization factors must be multiplied to the new velocities. The particles are split into gravitationally bound (where Φ + K ≤ 0) and unbound (where Φ + K > 0). The unbound particles new velocities are first multiplied by a random number in the range [0.8, 1.0] and subsequently normalized by multiplying them with q P P |Φ| Krand
After this normalization the kinetic energy is determined as 2 , Krand,norm = 21 vnew where vnew is the concatenation of initially bound, and the now bound particles (after normalization). Then the new cartesian velocities are normalized by multiplying them with q , where < Kinitial > and < Krand,norm > are the kinetic energies initially and after randomization plus normalizations,qrespectively. This gives the final velocities 2 2 2 vf inal = vx,f inal + vy,f inal + vz,f inal Now the kinetic energy is Kf inal = 12 vf2inal and the average value equals the average of the initial kinetic energy.
35
r Figure 19: Density profiles. log(ρ) vs. log r−2 for Sim. IIa of stable structures CS4 , CS5 , CS6 , DS1 and D2 . First subplot from the left shows the IC’s, middle subplot shows final products and right subplot shows final products of the control runs where no perturbations were applied to any of the structures. The right subplot thus serves as a efficient means of comparison with the final products of Sim. IIa. The formation of a core is seen for the final products as time evolves. 20 radial bins is used for these structures which have N = 105 particles.
36
Figure 20: β profiles for Sim. IIa of stable structures CS4 , CS5 , CS6 , DS1 and D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. Initial conditions with either complete velocity isotropy or slightly anisotropic ones are both seen to be driven towards a new attractor in the outer regions where they tend to land close to β = 0.1 (see middle panel). The control runs which are simulated with no perturbations interfering can be seen in the right panel. They show no significant departure from the initial conditions shown in the left panel.
37
Figure 21: γ profiles for Sim. IIa of stable structures CS4 , CS5 , CS6 , DS1 and D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. The final structures in the middle panel can be seen to get attracted towards a characteristic curve. The γ profiles thus have a preferred structure as very different initial conditions all flow towards this configuration. The control runs which are simulated with no perturbations interfering can be seen in the right panel. They show no significant departure from the initial conditions shown in the left panel.
38
Figure 22: κ profiles for Sim. IIa of stable structures CS4 , CS5 , CS6 , DS1 and D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. The middle panel reveals an attractor for the end products wrt. the κ-profiles. This attractor r is present for log r−2 > 0.5. The control runs which are simulated with no perturbations interfering can be seen in the right panel. They show no significant departure from the initial conditions shown in the left panel.
39
Figure 23: Attractor plot in the (β,γ)-space for Sim. IIa of stable structures CS4 , CS5 , CS6 , DS1 and D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. The left panel shows how the initial conditions spread out and fill up a large volume in the stable region of the Jeans parameter space spanned by β and γ. This makes the attractor very apparent in the middle panel where structures are driven towards β = 0.1 in the outer regions. Final products are shown together with a final product from a sim. of type I performed by S.H. Hansen and M. Sparre for comparison. Also shown is the linear β-γ relation discovered by Hansen and Moore (2006) for comparison. The pink upper right area is found to be an unstable region by An and Evans (2006).The control runs which are simulated with no perturbations interfering can be seen in the right panel. They show no significant departure from the initial conditions shown in the left panel.
40
Figure 24: Attractor plot in the (β,γ + κ)-space for Sim. IIa of stable structures CS4 , CS5 , CS6 , DS1 and D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. In this projection the attractor is still clearly seen. We still compare it with the results found by others. The control runs which are simulated with no perturbations interfering can be seen in the right panel. They show no significant departure from the initial conditions shown in the left panel.
41
6.0.3
IIb: Energy exchange (radial velocity perturbations)
Using structures CS4 and D2 , this simulation perturbs only the radial velocities of particles. Specifically, for all of these structures the total velocities inside spherical bins is perturbed for each of its particles by multiplying those given particles radial velocities (vr ) by random numbers in the range [0.7, 1.3] (corresponding to perturbing the kinetic energies by random numbers in the range 12 · [0.49, 1.69]), taken from the continuous uniform distribution. Following this exchange of kinetic energies between particles in fixed spherical bins, a normalization factor is applied to the total velocities, which ensures energy conservation. it has the same functional form as the one applied in sim. IIa, but will yield different numerical values as sim. IIb does not perturb all three spherical velocities but only the radial one. The normalization factor will thus be smaller in this sim. After these velocity kicks the structures are simulated under their self gravity allowing them to flow a bit under their new velocities. This procedure is then repeated for a total of 20 runs (with one initial run where there is no perturbation. This allows the OM models to reach equilibrium, as they are not perfectly equilibrated when set up as OM models. It is done for Edd structures as well to be certain of equilibrium before any perturbations take effect). After the first 10 runs, the random number range is unchanged but the runs 11 → 20 are given longer timespans (tsim = 500 corresponding to 5 dynamical times at r = 12.7rs ) to allow structures more flow. Starting from the Cartesian quantities (coordinates and velocities) one of the first steps in the algorithm used for the perturbations is to perform a transformation into spherical quantities. This can be done by applying a function F which maps real values from the cartesian domain R3 onto the spherical range R+ × [0, π] × [0, 2π). F has the components:
p x2 + y 2 + z 2 z θ = arccos( ) R y φ = arctan( ) x vR = sin(θ)cos(φ)vx + sin(θ)sin(φ)vy + cos(θ)vz R=
vθ = cos(θ)cos(φ)vx + cos(θ)sin(φ)vy − sin(θ)vz vφ = −sin(φ)vx + cos(φ)vy The coordinate transformation from Cartesian to spherical curvilinear coordinates has the following Jacobian matrix:
∂R ∂x JF (x, y, z) = ∂θ ∂x ∂φ ∂x
∂R ∂y ∂θ ∂y ∂φ ∂y
x ∂R ∂z R ∂θ = 0 ∂z ∂φ y − 2 ∂z x + y2
The determinant of this Jacobian matrix is 42
√ 1 R2 −z 2
y R 0
x2
x + y2
z R
1 −√ R2 − z 2 0
(41)
Now the spherical velocities are ready to be manipulated inside the number bins. Radial velocities are randomized as described above, and the speed is found as vtot = q 2 vR + R2 vθ2 + R2 vφ2 sin(θ)2 . Some of the particles might become gravitationally unbound after this procedure. This is checked and any such unbound particles are rebound by multiplying their new randomized radial velocity component √ by another random number in the range [0.8,1.0] and then a normalization factor of |Φ| Krand , where |Φ| is the numerical value of the gravitational potential inside any particular bin and Krand is the total kinetic energy of the unbound particles inside that bin. A final normalization factor is then applied to all particles inside anypgiven bin in order to obtain energy conservation. This factor is given by Kratio = < Kf inal >, where < Kinit > is the initial total kinetic energy of the particles inside that bin and < Kf inal > is the final total kinetic energy of the particles inside that bin. The operation to ensure energy conservation thus becomes: vR,f inal = vR,init · Kratio , where vR,init are the velocities prior to the perturbation. Final step of the perturbation algorithm is to transform spherical quantities back to Cartesian (so that the perturbed structure can be saved as a snapshot-file of correct format which the GADGET-2 code can then read and simulate). This can be done by applying another function F, which maps the spherical domain R+ × [0, π] × [0, 2π) onto the Cartesian range R3 . This new F giving the back-transformation has the components:
x = Rsin(θ)cos(φ) y = Rsin(φ)sin(θ) z = Rcos(θ) vx = sin(θ)cos(φ)vR + cos(φ)cos(θ)vθ − sin(φ)vφ vy = sin(φ)sin(θ)vR + sin(φ)cos(θ)vθ + cos(φ)vφ vz = cos(θ)vR − sin(θ)vθ The coordinate transformation from spherical to Cartesian coordinates has the following Jacobian matrix: ∂x ∂R JF (r, θ, φ) = ∂y ∂R ∂z ∂R
∂x ∂θ ∂y ∂θ ∂z ∂θ
The determinant of this 6.0.4
∂x sin(θ)cos(φ) Rcos(θ)cos(φ) −Rsin(θ)sin(φ) ∂φ ∂y = sin(θ)sin(φ) Rcos(θ)sin(φ) Rsin(θ)cos(φ) ∂φ ∂z cos(θ) −Rsin(θ) 0 ∂φ (42) Jacobian matrix is R2 sin(θ).
IIc: Energy exchange (tangential velocity perturbations)
Using structures CS4 and D2 , this simulation perturbs only the tangential velocities of particles. Specifically, for all of these structures the total velocity inside spherical bins is perturbed for each of its particles by multiplying those given particles tangential velocities vθ and vφ by random numbers in the range [0.9, 1.1] before each of the first ten simulations (each lasting for a duration of 1 tdyn at r = 12.7rs ) and random numbers in the range [0.7, 1.3] before each of the next twenty (for D2 ) or ten (for CS4 ) simulations (each of which this time 43
Figure 25: γ + κ vs. β for IC, final product and control run of sim. IIb, of stable structure CS4 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. No attractor is found. This agrees with work done by others [4]. The conclusion here must be then that not all perturbations will work when trying to reveal natures attractors. In [4] this is compared to a ball being continuously kicked uphill. The ball might wish to fall down (flow) but never get a chance to if it is repeatedly being kicked upwards (e.g. radially perturbed as in this case). Perhaps this experiment is just not in agreement with any physical process.
44
Figure 26: γ + κ vs. β for IC and final product of sim. IIb of structure D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. No attractor is found. This agrees with work done by others [4]. The conclusion here must be then that not all perturbations will work when trying to reveal natures attractors. In [4] this is compared to a ball being continuously kicked uphill. The ball might wish to fall down (flow) but never get a chance to if it is repeatedly being kicked upwards (e.g. radially perturbed as in this case). Perhaps this experiment is just not in agreement with any physical process.
45
lasting for a duration of 5 tdyn at r = 12.7rs , allowing structures more flow) (corresponding to perturbing the kinetic energies by random numbers in the range 12 · [0.49, 1.69]), taken from the continuous uniform distribution. Following this exchange of kinetic energies between particles in fixed spherical bins, a normalization factor is applied to the total velocities, which ensures energy conservation. it has the same functional form as the one applied in sim. IIa , but will yield different numerical values as sim. IIc does not perturb all three spherical velocities but only the two tangential ones. The normalization factor will thus be smaller in IIc than the one utilized in IIa but on average larger than the one in IIb . After these velocity kicks the structures are simulated under their self gravity allowing them to flow a bit under their new velocities. This procedure is then repeated for a total of 20 runs for CS4 and 30 runs for D2 (with one initial run where there is no perturbation. This allows the OM models to reach equilibrium, as they are not perfectly equilibrated when set up as OM models. It is done for Edd structures as well to be certain of equilibrium before any perturbations take effect). CS4 only gets 20 runs as it does not change significantly and thus already seem to start out at a stable point. D2 is however simulated for 30 runs at it changes significantly after each simulation. After 20 runs it has reached a stable point where the next 10 runs until run 30 has no real effect on D2 at all. Similarly to IIb , the IIc sim. transforms Cartesian into spherical quantities, divides the structure into particle number bins, perturbs and normalize velocities and energy, and finally transforms back to Cartesian form followed by saving into a new snapshotfile which can then be read and simulated by the GADGET-2 code. 6.0.5
IId: Energy exchange (speed perturbations)
Using structures CS4 and D2 , this simulation perturbs only the speed of particles and not the direction of the velocity vector. Specifically, for all of these structures the total velocities inside spherical bins is perturbed for each of its particles by multiplying those given particles cartesian velocities (vx , vy and vz ) by the same random numbers for each three directions (thus only changing the speed) in the range [0.7, 1.3] (corresponding to perturbing the kinetic energies by random numbers in the range 21 · [0.49, 1.69]), taken from the continuous uniform distribution. Following this exchange of kinetic energies between particles in fixed spherical bins, a normalization factor is applied to the total velocities, which ensures energy conservation. it has the same functional form as the one applied in sim. IIa. After these velocity kicks the structures are simulated under their self gravity allowing them to flow a bit under their new velocities. This procedure is then repeated for a total of 20 runs (with one initial run where there is no perturbation. This allows the OM models to reach equilibrium, as they are not perfectly equilibrated when set up as OM models. It is done for Edd structures as well to be certain of equilibrium before any perturbations take effect). After the first 10 runs, the random number range is unchanged but the runs 11 → 20 are given longer timespans (tsim = 500 corresponding to 5 dynamical times at r = 12.7rs ) to allow structures more flow.
We have so far seen the ICs plugged into sim. I and II and the outcome of this procedure. The next section highlights a few features from these sims. 46
Figure 27: γ + κ vs. β for IC, 10, final product and control run of sim. IIc of structure CS4 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. No change is seen at any part of this structure from β = 0.2 and upwards. The inner fluctuations where β is smaller can be attributed to numerical noise as the radii here moves close to the softening length of the sim. The lack of change is most probably that structure CS4 already starts out very close to the attractor and therefore remains here. When analyzing the shape of this structure up close it is seen to fulfill the attractor values already found in [4], namely (γ + κ) = −8β for smaller radii and (γ + κ) = −0.7 − 4β for larger radii.
47
Figure 28: γ + κ vs. β for IC, 10, 20, 30, final product and control run of sim. IIc of structure D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. The IC is seen to start out as isotropic (D2 is generated by Eddingtons inversion method) but then systematically exhibits flow toward the same attractor as seen present in the previous figure for the CS4 structure. This structure is given more simulation time than CS4 in order to follow its evolution more fully. It is indeed seen to move closer and closer to the characteristic s-shaped attractor curve found by others [4]. It is therefore safe to conclude that sim. IIc (perturbing tangential velocities) will drive structures towards the attractor previously found.
48
Figure 29: γ + κ vs. β for IC and final product of sim. IId of structure CS4 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. The attractor is present and can be seen for the final structure 20_013 which has been driven towards β = 0.1 in the outer regions. The pink upper right area is found to be an unstable region by An and Evans (2006).The control runs which are simulated with no perturbations interfering can be seen as well. They show no significant departure from the initial conditions.
49
Figure 30: γ + κ vs. β for IC and final product of sim. IId of structure D2 . Structures are cut off at a radius of 32 times the scale radius. 20 radial bins are used. The attractor is present and can be seen for the final structure 20_013 which has been driven towards β = 0.1 in the outer regions. The pink upper right area is found to be an unstable region by An and Evans (2006).The control runs which are simulated with no perturbations interfering can be seen as well. They show no significant departure from the initial conditions.
50
such as a particles trajectory when tracking it through subsequent snapshots during a simulation, effects found when zooming in on different radial domains, how softening length introduces numerical noise in the inner region leading us to make a inner cut and how outer parts of structures have insufficient time to reach equilibrium due to the large inter-particle distances leading us to make an outer cut. We thus obtain a valid range of trust where all structures can be safely analyzed. The next section concludes with a mention of OM models and their inability to reach any attractor, in good agreement with previous work done by others.
7 7.1
Statistical analysis and results Time evolution
In order to determine the necessary amount of perturbation → simulation repetitions the time-evolution is shown for two different structures, namely CS4 and D2 . they are tracked from ICs to final products with intermediate snapshots included as well (10_005, 20_005 and 30_005 ). Here the analysis is based on the type IIa sims. As can be seen from the figures, ten repetitions are sufficient as the structures at this point have become stable. Stable in this context means that continuing the perturbation → simulation experiment will not drastically change the profiles in any way (wrt. ρ, β etc.). To see the effect of kick and flow independently figure 19 shows structure CS4 at snapshot 19_005, then immediately after the next kick at snapshot 20_000 and finally after flow for a duration of one dynamical time at snapshot 20_005. In conclusion the flow is when changes to the structures become apparent but these changes were initiated at the kick so the two are co-dependent in order for this experiment to work. This teaches us about the scaling of both the size of the kicks and the number of dynamical times optimal for allowing the structures to settle into new stable configurations. As can be seen in the previous section under Controlled sim. II, dynamical times were set to either 1, 3 or 5. Kicks were either 5%, 20% or 30%.
7.2
Selection of radial domains
To study local properties of halos more clearly, four distinct regions of structures are plucked out. Radii corresponding to the four γ-values -1.5, -2.0, -2.5 and -3.0 are taken as central values in bins spanning a with where excactly 104 particles are included. The result of this localized analysis is that the central, majority of particles in structures are well behaved for all sim. in this work (I, IIa , IIb , IIc and IId ) but that the very inner parts are fluctuating due to the approach of gravitational softening length and similarly the very outer parts fluctuates due to insufficient simulation time (the dynamical time at large radii can be very large). See figure 34.
7.3
Cuts and range of trust
Figure 35-37 shows examples of how cuts are chosen. Noise in inner and outer regions determines the cutting points for inner and outer cuts and these cuts are performed for both the γ vs. log r plots, the κ vs. log r plots and the β vs. log r plots. Comparing these three parameter plots with cuts found for each, the most conservative overall inner and outer cuts are determined and applied to that particular structure. This is done for all structures, and the attractor plot (final products after sim. I plotted as γ + κ vs. β) can then be seen for the stable regions of all structures undergoing sim. I. This principle is then applied to all simulations (I, IIa , IIb , IIc and IId ). 51
Figure 31: γ vs. β for 19_005, 20_000 and 20_005 of sim. IIa of structure CS4. Also shown is the linear β-γ relation discovered by Hansen and Moore (2006) for comparison. The pink upper right area is found to be an unstable region by An and Evans (2006). This figure shows us the impact of perturbation and flow independently. Whereas the kick does not change the structure significantly on this type of plot, the subsequent simulation produces substantial alteration. It is thus the combination of both which serves to drive structures towards new equilibria just the same way as violent relaxation and phase mixing have synergestic effects on galaxies during late stages of mergers.
52
Figure 32: γ vs. β for IC, 10, 20, 30 and final product of sim. IIa of structure CS4. Also shown is the linear β-γ relation discovered by Hansen and Moore (2006) for comparison. The pink upper right area is found to be an unstable region by An and Evans (2006). Already after 10 runs the attractor state is present where β = 0.1 in the outer region.
53
Figure 33: γ vs. β for IC, 10, 20, 30 and final product of sim. IIa of structure D2. Also shown is the linear β-γ relation discovered by Hansen and Moore (2006) for comparison. The pink upper right area is found to be an unstable region by An and Evans (2006). Already after 10 runs the attractor state is present where β = 0.1 in the outer region.
54
Figure 34: γ vs. log r for final product of sim. I of structure A (48_009). The structure is initially cut off at a radius of Rlim = 32 · rs (corresponding to the log10 value of 1.5) Four γ-values of interest are chosen (indicated by the horizontal orange lines): γ = −1.5, -2, -2.5 and -3, which correspond to four different log r values, namely log(r−1.5 ), log(r−2 ), log(r−2.5 ) and log(r−3 ) respectively (indicated by the vertical orange lines). In particular γ = −2 (marked by a purple ring) is of interest for Hernquist structures as its corresponding log(r−2 )-value indicates the point where the density slope takes on its mean value. r−2 thus introduces a natural length unit which makes comparison between different Hernquist structures with for example different number of particles more meaningful.
55
Figure 35: β vs. log r for final product of sim. I of structure A (48_009). The structure is initially cut off at a radius of Rlim = 32 · rs (corresponding to the log10 value of 1.5) after which one additional (inner) cut is applied to the β-profile. This inner cut (shown in red and located at log r = −0.5) gets rid of Poisson noise due to the discrete nature of the N-body code, The remaining points on the β-profile thus only exist in the log r interval of [-0.5,1.5], namely the points with numbers [16;48].
56
Figure 36: γ vs. log r for final product of sim. I of structure A (48_009). The structure is initially cut off at a radius of Rlim = 32 · rs (corresponding to the log10 value of 1.5) after which two further cuts are applied to the γ-profile. The inner cut (shown in red and located at log r = −0.5) gets rid of Poisson noise due to the discrete nature of the N-body code, whereas the outer cut (shown in blue and located at log r = 1.0) removes points far away from the halo center, where the particles have not yet had sufficient time to reach a stable equilibrium. The remaining points on the γ-profile thus only exist in the log r interval of [-0.5,1.0], namely the points with numbers [15;39].
57
Figure 37: κ vs. log r for final product of sim. I of structure A (48_009). The structure is initially cut off at a radius of Rlim = 32 · rs (corresponding to the log10 value of 1.5) after which one additional (inner) cut is applied to the κ-profile. This inner cut (shown in red and located at log r ≈ −0.9) gets rid of Poisson noise due to the discrete nature of the N-body code, The remaining points on the κ-profile thus only exist in the log r interval of [-0.9,1.5], namely the points with numbers [8;46].
58
r Figure 38: log(σ 2 ) vs. log( r−2 ) for one of the final products (198_093) of Sim. I for structure B. 20 bins logarithmic in radius is used and the structure is cut off at radius n P Rlimit = 104 in arbitrary units of radius. σ 2 is calculated as σ 2 = n1 · (vi − < v >)2 i=1
inside each bin, where n is the number of particles inside a particular bin and is the average velocity inside this bin. For the inner and middle part of this structure the total dispersion is much greater than the radial dispersion and greater than the tangential r dispersion, but at large radii (from log( r−2 ) ≈ 2.3) it can be seen that the radial part starts to dominate over the tangential part and finally almost the entire total dispersion comes from the radial dispersion whereas the tangential dispersion in the outer part is almost negligible.
59
7.4
Unstable OM models
OM models with HQ density profiles are unstable when ra < 1.33, since this will cause radial orbit instability (ROI), creating a central bar structure. Several such structures are created (C1 → C6 ) plus an additional OM structure with inner and outer density slopes (0,-5), (D1 ). None of them lands on an attractor in the Jeans parameter space. This is in agreement with [8]. See the previous section ’Overview of structures’ for more details concerning these profiles. We have highlighted various features from the simulations in this section and the next part summarizes the most important results for the entire project.
8
Conclusion
The γ + κ vs. β-plots gives useful insight to the effects of each different simulation types for various ICs. Sim. I clearly reproduced the attractor found earlier [7]. The main difference between this sim. and the one done by others is the increased number of perturbation->simulation repetitions as well as the removal of the few gravitationally unbound particles residing in the outer regions of structures toward the later stages of the simulations. The more numerous experiments done in this way indicates a true attractor which is robust against repeated perturbations->simultions as any true attractor must be (as well as any stable points). End products of sim. IIa landed on a almost vertical curve indicating that this type of sim. drives structures closer to isotropy. This behavior is in agreement with previous results [6]. The structures does not become completely isotropic but are attracted towards a state in the Jeans parameter space where β = 0.1 in the outer parts. This thus indicate the presence of a new attractor. The reason for this probably resides in the fact that this sim. type is symmetric in velocity kicks whereas previous work had been symmetric in energy kicks (perturbing kinetic energy in the range [0.25,1.75]) [4]. This subtle difference might lead to differences in the VDFs which might then dictate the outcome. Sim. IIb does not result in any attractor, as the parturbations and flow have no systematic effect on structures and they are seen to fill up a larger and larger span of the parameter space between β, γ and κ. Perhaps this is just an artificial type of simulation which does not accurately resemble real physical procecces in the universe. One could try to find systematic features in the flow by providing smaller velocity kicks than the ones here utilized. This is in agreement with [4]. End products of sim. IIc landed on the characteristic s-shaped attractor curve. This is in agreement with [4]. End products of sim. IId landed on the same almost vertical curve as the one found in IIa indicating that this type of sim. similarly drives structures closer to isotropy. This is the same outcome as the type IIa sim., which is to be expected as it is basically a subclass of IIa . This behavior (approaching isotropy) is in agreement with previous results [6]. Again the new attractor seems present in the outer parts of the end products.
60
A relationship between the velocity anisotropy, the radial slope of the density profile, and the radial slope of the squared velocity dispersion is thus confirmed for stable configurations when undergoing simulations of type I (G-perturbations) and IIc (perturbing tangential velocities). This relationship exists in the form of a one-dimensional attractor (s-shaped curve). concluding the side projects (See appendix B), cosmological simulations The cosmological analysis of dark halos morphology indicates that halo shape generally has slight departures from sphericity and can be either oblate or prolate in cosmological simulations. VDF Velocity profiles (or Velocity Distribution Functions) of dark matter halos are seen to depart slightly from Gaussian distributions and to be better fitted by Tsallis power laws (Generalized Gaussian distributions) which are more steep and have flatter tails. LOS A comparison is made between 3D structures and their corresponding line-of-sight (LOS) appearance showing that details in fine-structure is lost when going from the former to the latter. This is a complication related to observations that does not exist for simulations where a complete 3D view can be obtained. Bumpy road to universalities The radial slope of the density profile is seen to converge towards a characteristic shape with 6 extrema, i.e. 3 maxima and 3 minima. To analyse these features in more detail, each extremum can be applied in turn as a normalization to various structures to see their agreement with, or departure from this, in a closer look. All-in-all an attractor is found alongside other universal trends and it is therefore concluded that the results of structure formation and evolution is not completely random but follow deep underlying physical principles. There must be some physical reason behind this type of behavior as it has been found for a large span of ICs undergoing different types of sims. resembling physical processes taking place during galaxy mergers. The velocity distributions of DM needs to be understood in more detail or understanding the effects of violent relaxation and phase mixing better might give new clues. Hansen, Juncher and Sparre (2010) suggests that this attractor is fundamental for spherical dark matter structures and that it might be seen as a fundamental relation for dark matter; just as the NFW-profile (maybe the attractor can help in explaning why the NFW profile is universal.) and the σρ3 -relation. The attractor is at least one more interesting feature of our universe concerning dark matter, which might lead to new understanding in the cosmological context. Putting some of the pieces together, we can get closer to solving the Jeans eq.; The density is now modelled, and we have the knowledge of the pseudo-phase-space density as 2 (r). In order to deterwell. Putting these two pieces of information together, this fixes σrad 2 mine β it still remains to find the tangential velocity dispersion, σtan (r). Non-cosmological studies has been conducted to explore the properties of β-profiles further,e.g. searching for a linear relation between β and γ, which is found in the inner region of cosmological halos. Mass estimates of large structures using the J.E. (derived for a steady spherical system without bulk flow) can be off by up to 40 % when falsely assuming β = 0 in some cases. This M-β degeneracy (the mass will be different for different choices of β) could be solved
61
if β is known. So for dwarf galaxies for example if σr and ρ is found then after applying the attractor curve β will be known as well thus potentially eliminating the M-β degeneracy completely. The J.E. thus effectively removes one degree of freedom from the J.E. The advancements in thermodynamics driven by Maxwell and Boltzmann gave a microscopic understanding of universal trends between macroscopic quantities such as pressure, temperature and density for a ideal gas, namely by P = T · ρ which lead to the invention of modern statistical mechanics. In terms of DM, finding the relationships between the macroscopic quantities such as density ρ, velocity dispersion squared σ 2 and the velocity anisotropy parameter β is where we are at today. Searching for the underlying microscopic physics in terms of VDF’s (radial and tangential) will be one of the next steps needed to obtain a more fundamental understanding. In order to be certain if a configuration really represents a true attractor the eigenvalues related to this configuration must be considered. An attractor can have both positive and negative eigenvalues. They are called Lyapunov exponents since we follow them ’around in space’ (in this case the 3D Jeans parameter space. They are not calculated in a point). This is a possible next step to test attractors further. In [15] the flow towards the attractor found in [4] is examined and it is found that the further away ICs are from this attractor, the faster they will flow towards it. Also the larger the perturbations the larger the flow rate towards the attractor. Structures then slows down as they approach the attractor and are never seen to cross it (in a way that stars would when flowing towards the main sequence attractor found on a HR diagram) thus indicating a damped oscillator type system which could potentially be described by linear equations from which the Lyaponov exponents might be found and examined. It still remains to find the attractor in cosmological halos, which might have to do with the fact that β is different along different directions as well as structures not being totally spherical [9]. To see my analysis codes, please visit the following web-adress: https://bitbucket.org/dark_knights/darkmatterproject.
References [1] Martin Sparre Eddington-Code developed for Sparre & Hansen 2012, JCAP, 07,042 and Sparre & Hansen 2012, JCAP,10,049, available at https://github.com/martinsparre/Eddington-Code. [2] Martin Sparre Osipkov-Merritt-Code developed for Sparre & Hansen 2012, JCAP, 07,042 and Sparre & Hansen 2012, JCAP,10,049, available at https://github.com/martinsparre/OsipkovMerritt. [3] J. Binney and S. Tremaine, Galactic Dynamics: Second Edition, Princeton University Press, Princeton U.S.A. (2008). [4] S.H. Hansen, D. Juncher and M. Sparre, An attractor for dark matter structures, arXiv:1005.1643 [INSPIRE]. [5] S.H. Hansen and B. Moore, A Universal density slope - velocity anisotropy relation for relaxed structures, New Astron. 11 (2006) 333 [astro-ph/0411473] [INSPIRE]. [6] S.H. Hansen and J. Stadel, The velocity anisotropy-density slope relation, JCAP 05 (2006) 014 [astro-ph/0510656] [INSPIRE].
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[7] S.H. Hansen and M. Sparre, The behaviour of shape and velocity anisotropy in dark matter haloes, arXiv:1210.2392 [astro-ph.CO]. [8] A. Meza and N. Zamorano, Numerical stability of a family of Osipkov-Merrit models, arXiv:astro-ph/9707004 [9] M. Sparre and S.H. Hansen, Asymmetric velocity anisotropies in remnants of collisionless mergers, arXiv:1205.1799 [astro-ph.CO] [10] V. Springel, The Cosmological simulation code GADGET-2, Mon. Not. Roy. Astron. Soc. 364 (2005) 1105 [astro-ph/0505010] [INSPIRE]. [11] V. Springel, High perfomance computing and numerical modeling, Lecture 1: Collisional and collisionless N-body dynamics, 43rd Saas-Fee Course, Star formation in galaxy evolution: connecting numerical models to reality (11-16 March 2013) [12] M. Vogelsberger et. al., Dwarf galaxies in CDM and SIDM with baryons: observational probes of the nature of dark matter, arXiv: 1405.5216 [13] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, 1972. [14] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity. [15] J.A. Barber, H. Zhao, X. Wu and S.H. Hansen, Stirring N-body systems: Universality of end states, arXiv:1204.2764 [INSPIRE]. [16] B. Ryden, Introduction to Cosmology. [17] S.H. Hansen, Dark matter density profiles from the jeans equation, Mon. Not. Roy. Astron. Soc. 352 (2004) L41 [astro-ph/0405371] [INSPIRE]. [18] S.H. Hansen, Might we eventually understand the origin of the dark matter velocity anisotropy?, Astrophys. J. 694 (2009) 1250 [arXiv:0812.1048] [INSPIRE]. [19] S.H. Hansen and M. Sparre, A derivation of (half) the dark matter distribution function, arXiv:1206.5306 [INSPIRE]. [20] A.S. Eddington, The distribution of stars in globular clusters, Mon. Not. Roy. Astron. Soc. 76 (1916) 572. [21] L. Hernquist, An analytical model for spherical galaxies and bulges, Astrophys. J. 356 (1990) 359. [22] Power et. al., The inner structure of ΛCDM haloes - I. A numerical convergence study (2003) [23] G. C. Rasmussen, Computational Astrophysics: Structures of dark matter halos around galaxy clusters, exam report for the course ’Computational Astrophysics’ at NBI, UCPH (2015).
63
A
A note on simulation time
Most runs of sim. I and all runs of sim. II (for all structures and with Run 0 included) lasts 100 · T imeM ax, corresponding to one dynamical time tdyn at a radius r = 12.7 · rs . 1 Proof (assuming a HQ profile with ρo = 2π ): The dynamical time is given by 1 1 tdyn = √ =r G¯ ρ (r) G· M 4 πr3
(43)
3
setting G = 1 and inserting the mass-function for a HQ structure, M (r) =
rs ·r2 : (1+ rr )2 s
1
=
s
rs ·r 2 (1+ rr )2 s 4 πr3 3
v u u t
4 3 3 πr 2 rs ·r (1+ rr )2
s
4 3 3 πr
=
s
· (1 +
r 2 rs )
rs · r2 s
4 3 πr
· (1 +
=
r 2 rs )
rs and this will equal 100 when s
4 3 πr
· (1 +
r 2 rs )
rs 4 3 πr
· (1 +
r 2 rs )
rs r · (1 + rrs )2 rs
=
= 100 ↔
= 104 ↔ 3 · 104 4π
substituting r = x · rs we find x(1 + x)2 =
3 · 104 ↔ 4π
x ' 12.7 and so 100 simulation times (TimeMax = 100) equals one dynamical time tdyn , at r = 12.7 · rs .
B B.1
Side projects Cosmological SIMS
Using the Gadget-2 code, two cosmological simulations with 1283 and 2563 dark matter particles were run in a cosmological box with a co-moving volume of (10 Mpc)3 . Gravitational softenings of 7.4 and 1.8 kpc were used, respectively. The simulations were initiated at z = 63, and the initial conditions 64
Figure 39: Spherical coordinates r, θ and φ. The 3 boldtype coordinate axes represent the Cartesian axes x, y and z. The grey areas shows two small surface areas of concentric spheres around Origo, which together define a small volume element in configuration space. With r ≥ 0, θ ∈ [0; π] and φ ∈ [0; 2π] we can define any point in space. credit: http://pleasemakeanote.blogspot.dk/2010/02/9-derivation-of-continuity-equation-in.html were created with the public N-GenIC code, available at the Gadget website: http://www.mpa-garching.mpg.de/gadget/
B.2
Velocity distribution functions
Velocity distribution functions (VDF’s) are fundamentally important in order to understand the behavior of DM dynamics. The initial velocity distributions are Gaussian and have velocity dispersions, σrad and σtan , which I later use in the analysis code (they can be found from the Jeans equation, see previous section). It is worth mentioning that in reality, the velocity distributions is probably not Gaussian but the velocity is anisotropic. The Dark Matter velocity is badly understood, however this will normally not effect the discovery of dark matter in direct experiments. But it is critically important to understand the DM velocity better in order to compare different experiments with each other, as well as comparing direct detections with colliders. So it is crucial to develop a better understanding in order to detect the actual DM particles. The total velocity of each particle is decomposed into the 1D radial velocity, and the 2D tangential velocity which can be further decomposed into two mutually orthogonal velocity vectors residing in the tangential plane of the halo wrt the radius vector. They are vθ and vφ . From the cartesian coordinates, the spherical coordinates can be found:
65
p x2 + y 2 + z 2 z θ = arccos |r| y φ = arctan x r=
(44)
Similarly, x = rsinθcosφ (45)
y = rsinθsinφ z = rcosθ
The total velocity vector can be expressed in terms of Cartesian unit vectors, ˆi, ˆj and ˆ k, as vx (46) v = vy = vxˆi + vy ˆj + vz kˆ vz ˆ as Or wrt the spherical unit vectors , rˆ, θˆ and φ, r˙ ˙ v = rθ˙ = rˆ ˙ r + rθ˙θˆ + rφsinθ φˆ ˙ rφsinθ
(47)
So the radial velocity becomes x · vx + y · vy + z · vz r·v = = |r| |r| |r| · sinθcosφ · vx + |r| · sinθsinφ · vy + |r| · cosθ · vz + = |r| sin(θ)cos(φ)vx + sin(θ)sin(φ)vy + cos(θ)vz
vr =
(48)
the tangential velocity component vθ is vθ = d dθ
dr dθ
·v = |r|
· (rsinθcosφ) · vx +
d dθ
· (rsinθsinφ) · vy + |r| cos(θ)cos(φ)vx + cos(θ)sin(φ)vy + sin(θ)vz
d dθ
· (rcosθ) · vz
=
(49)
the tangential velocity component vφ is vφ = d dφ
dr dφ
·v
|r|
=
· (rsinθcosφ) · vx +
d dφ
· (rsinθsinφ) · vy + |r|
d dφ
· (z) · vz
=
(50)
d d · (cosφ) · vx + · (sinφ) · vy + 0 = dφ dφ − sin(φ)vx + cos(φ)vy , since for the φ-plane, z = 0, θ = 90 deg and so sinθ = 1. and finally the 2D tangential speed is q |vtan | = vθ2 + vφ2 (51) 66
Figure 40: A standard normal distribution, scaled appropriately for easier comparison with the following velocity histograms.
Figure 41: FWHM and σ In order to better compare different radial bins, each velocity or speed is normalized by dividing them with their corresponding velocity dispersions. These new dimensionless normalized velocities and speeds are denoted by u; The radial normalized velocity is ur =
vr σr
(52)
and the tangential normalized speed is q ut = uθ 2 + uφ 2 =
s
vθ σθ
2
+
vφ σφ
2
(53)
To see how close the VDFs are to being Gaussian, from the plots we can determine the √ FWHM and use the relation F W HM = 2 · 2ln2σ ≈ 2.355σ, where σ is the velocity dispersion. This value of σ can then be compared to the real σ that is computed directly. σ(F W HM ) The ratio σ(computed) · 100% then gives the offset in pct. 2 = A word on the dispersions: The variance of tangential velocities are defined as σtan σθ2 + σφ2 , but this only applies when the angular momentum is the same for all directions wrt the halo centre. That is easily checked in Python as e.g. 67
2
Figure 42: a · e−b·x and f(q) for q =
5 3
and q = 12 .
Lx = np.cross(r, vx ) Ly = np.cross(r, vy )
(54)
Lz = np.cross(r, vz ) These quantities are found to be identical. For a spherical system of gas particles, the radial VDF would read vr 2 (55) f (vr ) = exp(− 2 ) 2σr and the tangential VDF would be f (vt ) = vt · exp(−
vt 2 ) 2σt2
(56)
The DM particle velocities is compared with two different fits: a Gaussian function 2 a · e−b·x and a more general power-law fitting function, f(q). Figure 1 shows a comparison of a standard normal distribution function with f(q): 2 f(q) and ae−bx are both good fits to vtan around γ = −2.0, which is seen in figure 2:
68
Figure 43: vtan divided by σtan , for 4 different radial bins. Shown in lin-lin, lin-log, log-lin and log-log
Figure 44: vtan together with Gaussian and Tsallis fit.
69
Figure 45: vtan divided by Gaussian function.
Figure 46: Histograms of the ratio ut = σvtan for 5 different snapshots (this makes it easier tan to compare different radial bins than it would be for vtan alone). Data for 4 different radial bins are shown, each containing 104 particles and centered at the radii where γ = −1.5, −2.0, −2.5 and −3.0 respectively.
70
Figure 47: vr , vθ and vφ divided by σr , σθ and σφ respectively, for 4 different radial bins. Shown in lin-lin, lin-log, log-lin and log-log
Figure 48: vr , vθ and vφ together with Gaussian and Tsallis fit.
71
Figure 49: vr , vθ and vφ divided by Gaussian function.
B.3
Line-of-sight overdensities
Starting with an end-product from a G-perturbation simulation, which is no longer perfectly spherical and therefore more closely resembles a cosmological DM halo, a plot of radial velocity, vr vs. radius, r (for a 3D simulation file) is compared to the corresponding plot of velocity along the x-direction, vx (taken to be the line-of-sight, LOS) vs the p projected radius, R (given by R = y 2 + z 2 ). It is possible to simulate 3D structures, but in reality it is only possible to observe the LOS-velocity, vLOS here defined to be vx . R is the projected radius onto LOS and r is here the 3D radius vector from the simulation. An interesting question to investigate is: How does overdensities in 3D look in this corresponding 2D representation (vx vs R)? The biggest difference between the r, vr -graph and the R, vx -graph is the amount of information lost when going from 3D quantities to line-of-sight quantities. Notice in particular the high degree of substructure in the r, vr -graph; we see 5 arms corresponding to each variation of G, which is in a way frozen into the phase-space volume. When G is increased, the radial velocity grows rapidly for larger radii, and when G is decreased, the radial velocity falls off just as rapid. This figure shows the inner part (up to radius 50) of the phase-space volume for particles in the file Hernquist10000_G1.0_10_009. If we pay close attention to the zero-line on the y-axis, in general we find as many particles over as under this line in the most central part up until about radius 20. The central part is thus symmetric around the zero line. At higher radii there tend to be more particles with positive radial velocities which are the ones that has escaped the structure and will continue heading outwards forever. This means that the central phase-space volume of particles are in equilibrium, but the outer part is not. The outer particles below the horizontal zero-line are heading towards the central phase-space volume again. This is particles experiencing a bit of infall. We can thus track the particles which are no longer bound.
72
Figure 50: For the file Hernquist10000_G1.0_10_009 containing 106 particles we here see a comparison of the radial velocity as function of radius (left panel) to the LOS-velocity vs the projected radius for the G-perturbation simulation. The left panel showing the radial velocity clearly portrays the presence of substructure which is washed out in the corresponding line-of-sight plot in the right plot. This point makes it clear that numerical simulations can provide a full picture absent when performing observations naturally restricted to the line of sight. The radial velocity plot has 5 distinct filaments, one for each of the perturbations. The filament furthest to the right corresponds to the first time particles escapes the structure.
Figure 51: For the file Hernquist10000_G1.0_10_009 containing 106 particles we here see a comparison of the radial velocity as function of logarithmic radius (left panel) to the LOS-velocity vs the logarithmic projected radius. This gives a great resolution of the center.
73
Figure 52: For the file Hernquist10000_G1.0_10_009 we here see a comparison of the radial velocity as function of radius and logarithmic radius (left panel) to the LOS-velocity vs the projected radius and logarithmic projected radius. A thousand particles have been picked out for this analysis and the structure is cut off at radius = 50. Looking at the top left subplot, basically the same number of particles seems to be present both above and below the zero-line at inner and middle-regions. at r > 40 positive radial velocities starts to dominate over negative ones indicating a small particle flow away from the structure at large radii. This is expected as outer particles are less bound and might not have had sufficient time to reach equilibrium.
74
Figure 53: The final products for the simulations B, C4 , C5 , C6 , D1 and D2 . Here σr2 vs. logarithmic radius and radial velocity vs the logarithmic radius is shown. N = 105 particles are used for this analysis and the structure is cut off at radius = 10000.
B.4
The bumpy road to universalities
From the top panel of figure 51 there appear to be some overall trends to σr2 : in the inner part, Sim B clearly has the largest σr2 , which then becomes smaller than the other simulations σr2 in the outer part. From the bottom panel of figure 51 it is seen that on average there is the same amount of radial velocities above and below the zero line at inner and middle regions. This indicates the structures are in equilibria here and their particles are gravitationally bound. The outer region has an increase in radial velocities due to a few unbound particles which dominate the picture here. These have later been removed from the structures which then become flattened in this type of plot even for outer regions.
C
Functional analysis
Functional analysis of the β and γ functions are performed graphically together with a presentation of the restriction β < − γ2 which has to be met in order to obtain stable structures (An & Evans 2006). A structure which lie in the region where β > − γ2 might be created, but as soon as the time integration begins the structure will no longer be stable. This unstable region is shown in figure 19 as a pink region in the upper right corners of each subplot. It has been found to be unstable by An & Evans in 2006;i.e. for a cusped density profile which goes as rγ near the center, the limiting value of the anisotropy parameter β cannot be greater than − γ2 at the center. It follows from the non-negativity of the phase-space density. By tuning the two parameters ra (anisotropy radius) and rs (scale radius) it is possible to approach the unstable region without ever crossing into it. This knowledge is helpful when choosing how to set up the initial structure; If one wishes to create a stable structure close to or far away from the unstable region one simply has to make the corresponding choices for these two parameters. We learn from figure 19 that a small value of ra , a large value 75
Figure 54: IC and final products for the simulations B, C4 , C5 , C6 , D1 and D2 . The final products show universal trends: there seem to be three distinct local minima and similarly three distinct local maxima to the γ profiles at the outer volume (where log r > 1) of the various structures. For D1 and D2 the extrema almost overlap, which is also the case for C4 , C5 and C6 . B has unique positions of extrema.
Figure 55: Final products for the simulations C4 , C5 , C6 , D1 and D2 . Here the individual γ profiles are shown vs. log( dr3 ), where d3 are the γ-value of each profile at its third local minima.
76
Figure 56: Time evolution of γ-profile for the final states of the unstable structure D1 (left panel) and the final states of the stable structure D2 (right panel) from sim. I. From the top and downwards the number of radial bins is 20, 50, 100 and 200 respectively. The final states are 48_093 and 49_093 for both structures. From plots like these it is concluded that the optimum number of radial bins for structures with N = 105 particles is 20 and that the optimum number of radial bins for structures with N = 106 particles is 50.
Figure 57: Final products for the simulations B, C4 , C5 , C6 , D1 and D2 . Here the radial velocities divided by the radial velocity dispersions are shown vs. r and log r, for both 50 and 20 radial bins.
77
Figure 58: Zoom in on previous figure. Due to Poisson noise, the regions logr < -1 and logr > 2 can not be trusted. The vertical axis showing ur has also been narrowed down to a smaller range for more clarity. of rs can bring our structure closer to the unstable region. Notice how these effects are unaffected by our choice of ρ0 as it is a constant that will not affect the differentiation performed when computing the γ array. But in practical purposes it is best to make some compromise to avoid too large of a computational cost (longer simulation time). Also, there is no guarantee that a structure outside the unstable region is stable. This has to be checked. In fact the contrary might be true as well; It might be possible to create a stable structure that crosses into the pink region and out again, since the An & Evans paper from 2006 only concludes this instability based on a smooth γ-profile. For all structures, the characteristic radius is rs = 1, and the normalization constant ρ0 has been chosen so that the total mass inside r = 13rs is 1. This ensures that the dynamical time mentioned in the previous section , for particles inside r = 13rs is smaller than 100 time units (which is the duration of all simulations performed). For the Osipkov-Merritt model, we have seen in the previous section that the β profile is β=
r2 r2 + ra2
(57)
The anisotropy radius ra is varied from 1.2rs to 0.2rs . When ra exceeds rs instabilities can be avoided. It is shown (Kazantzidis, Stelios) that ra ≥ 1.33rs will ensure stability. In the setup the particle number is varied from 104 to 106 .
D D.1
Brief overview and description of analysis codes VDFs
The analysis is done with three separate programs which I have written in the Python programming language. Amongst other things they agree with cosmological simulations about virial radius which tend to be rvir = 10r−2 , with r−2 = r2s (for a Hernquist profile). For a NFW profile, rs = r−2 . To see the programs, look under appendices D, E and F, where there is also a short introduction to each program. Comment lines are included into 78
Figure 59: Analysis of the β and γ functional expressions for different choices of scale 1 radius and anisotropy radius. The ρ0 normalization parameter is kept fixed at ρ0 = 2π . γ Plotted together with the unstable region where β > − 2 . This figure gives a qualititive idea of the kind of structures that might fall into the stable category.
Figure 60: 2D view of IC for structure A. we first see the particles y-positions vs. their x-positions and then their z-positions vs. their x-positions. The halo is cut off at a radius of Rlim = 32.
79
the code, to clarify each step taken. In this section I will simply attempt to highlight the most prominent features of each of these codes. For the first code, ’Read.py’, the central part is the following for-loop which we shall now investigate further: 1
# Divide the
2
for i in range (0, int( nr_binning_bins )):
structure
t h e number o f 3
into
logarithmic
radial
bins # loop over
bins
min_R_bin_i = binning_arr_lin_log10 [i]
# start
of
bin 4 5
max_R_bin_i = binning_arr_lin_log10 [i+1] # e n d o f b i n posR_par_inside_bin_i = np.where ((R> min_R_bin_i ) & (R< max_R_bin_i ))[0] # p o s i t i o n o f p a r t i c l e s i n s i d e a r a d i a l
6
bin # number o f
7
nr_par_inside_bin_i
8
# Volume o f
9
10 11
12 13 14 15
particles
inside a radial
bin :
= len( posR_par_inside_bin_i )
cluster : Volume_cl = (4./3.) ∗ np.pi ∗ ( max_R_bin_i ∗ ∗ 3 − min_R_bin_i ∗∗ 3) den_cl = nr_par_inside_bin_i / Volume_cl # n u m b e r d e n s i t y rho_cl = m ∗ den_cl # d e n s i t y (m i s t h e m a s s o f e a c h p a r t i c l e , m = M/N = 1 /N) # save to l i s t s
density_arr . append ( den_cl ) Volume_arr . append ( Volume_cl ) Invers_Volume_arr = np.log10(np. divide (np.ones(len( Volume_arr )),Volume_arr )) After cutting the cluster structure up into logarithmic linear radial bins, each bin contains a certain number of particles whose various physical quantities is then averaged,e.g. the number density given by number of particles in each bin divided by the volume of this spherical shell. The number of particles for a certain bin, i, is found by taking the length of the array containing all particles inside bin_i. The volume of a spherical shell in general is just the volume of the remainder of the whole sphere´s volume after subtracting the volume of the inside sphere (with radius equal to the start value of the bin radius). This is then used in the line above which writes
1
Volume_cl = (4./3.) ∗ np.pi ∗ ( max_R_bin_i ∗ ∗ 3 − min_R_bin_i ∗∗ 3) So now the number density is ready to be plotted, and can be fitted with a Hernquist density profile. (for comparison I also fitted with a NFW-profile). In the second analysis code, ’Sigma.py’, the most central aspect is firstly to find the velocity dispersions and secondly to compute the β, κ and γ variables. Let us first see how the velocity dispersions are found. Basically it is an expansion of the previous for-loop used to find the density.
1
for i in range( nr_binning_bins ): number o f
2
# loop over the
bins
min_R_bin_i = binning_arr_lin_log10 [i]
# start
of
bin 3
max_R_bin_i = binning_arr_lin_log10 [i+1]
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# end o f b i n
4
5
posR_par_inside_bin_i = np.where (( R_hob_par > min_R_bin_i ) & (R_hob_par < max_R_bin_i )) # P a r t i c l e p o s i t i o n s nr_par_inside_bin_i = len( posR_par_inside_bin_i [0]) # number o f
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10 11
14
15 16
17 18
inside a radial
bin
if nr_par_inside_bin_i == 0: continue v2_inside_bin_i = v2[ posR_par_inside_bin_i ] sigma2_inside_bin_i = (1./( nr_par_inside_bin_i +1.)) ∗ np. sum( v2_inside_bin_i ) sigma2_arr . append ( sigma2_inside_bin_i ) # s i g m a 2 t o t a l bin_radius_arr . append (( max_R_bin_i + min_R_bin_i )/2) # sigmarad2
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particles
radial
vrad2_inside_bin_i = v_r[ posR_par_inside_bin_i ] ∗∗ 2 sigmarad2_inside_bin_i = (1./( nr_par_inside_bin_i +1.)) ∗ np.sum( vrad2_inside_bin_i ) sigmarad2_arr . append ( sigmarad2_inside_bin_i ) Volume_cl = (4./3.) ∗ np.pi ∗ ( max_R_bin_i ∗ ∗ 3 − min_R_bin_i ∗∗ 3) # c l u s t e r v o l u m e den_cl = nr_par_inside_bin_i / Volume_cl # n u m b e r d e n s i t y rho_cl = m ∗ den_cl # d e n s i t y (m i s t h e m a s s o f e a c h p a r t i c l e , m = M/N = 1 /N)
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density_arr . append ( den_cl ) # save array Volume_arr . append ( Volume_cl ) # s a v e a r r a y sigma2_arr = np.array( sigma2_arr ) # square of total
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velocity
dispersion
sigmarad2_arr = np.array( sigmarad2_arr ) bin_radius_arr = np.array( bin_radius_arr ) Notice the way the total velocity dispersion, σ 2 , is found by dividing the sum of the square of the velocities by the number of particles. Normally the sum would contain the difference of the velocities and the mean velocity, squared, but the mean velocity is already accounted for by defining new velocities, in order to make this loop simpler. The radial velocity dispersion , sigmarad2, is found in an analogous way, by dividing the sum of the square of the radial velocities by the number of particles. The tangential velocity dispersion is then just a question of subtracting the radial from the total velocity dispersion, and we have all three. They can now be used to compute the β, κ and γ variables:
1
# kappa
2
for
3 4 5 6
7
8 9 10 11 12
i in range(len( sigma2_arr )): if i == 0 or i == len( sigma2_arr ) −1: kappa_arr . append (np.nan) continue dlogr = np.log10( bin_radius_arr [i+1]) − np. log10( bin_radius_arr [i −1]) dlogsigmarad2 = np.log10( sigmarad2_arr [i+1]) − np. log10( sigmarad2_arr [i −1]) kappa_arr . append ( dlogsigmarad2 /dlogr)
# gamma
for
i in range(len( density_arr )): if i == 0 or i == len( sigma2_arr ) −1: gamma_arr . append (np.nan) 81
(a) High resolution
(b) Low resolution
Figure 61: Image of the 10 Megaparsec structure from the Gadget-2 simulation with Gperturbations. The most luminous clumps in the structure are galaxy cluster halos of dark matter. 13 14
15
16 17
continue dlogr = np.log10( bin_radius_arr [i+1]) − np.log10( bin_radius_arr [i −1]) dlogrho = np.log10( density_arr [i+1]) − np.log10( density_arr [i −1]) gamma_arr . append ( dlogrho /dlogr) # calculate
sigmatheta
18
sigmatheta2_arr = ( sigma2_arr − sigmarad2_arr )/2.
19
# calculate
20
beta
beta
_arr = 1. − sigmatheta2_arr / sigmarad2_arr
2 The strategy employed here to find κ is first finding d log r, then d log σrad and their
ratio,
2 d log σrad d log r .
Very similar for γ, d log r is found, then d log ρ, and then taking the ratio σ2
ρ θ between the two, dd log . This is done for all datasets, 2 log r . Now for β, it´s simply β ≡ 1 − σrad plotted, and saved as text files to be used in the following. Third and final analysis program, ’gamma_kappa_beta.py’ now takes the text files from previous code, and overplots them. This shows clear indication of a preférred structure in (β,γ,κ)-space, see conclusion in next section.
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Figure 62: From the previous high resolution image of the 10 Mpc structure, the largest galaxy cluster halo of dark matter has been cut out to produce this image.
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Figure 63: Initial plot of G-perturbations with β = 0 for a Hernquist structure (0G00_IC_000.hdf5).
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Figure 64: Final plots of G-perturbations with β = 0 for a Hernquist structure (0G20_Final_000.hdf5). Notice the structure has expanded significantly with respect to the initial structure.
(a) The whole structure
(b) Zoom-in on cluster
Figure 65: Initial plots of Osipkov-Merritt structure before G-perturbations.
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(a) The whole structure
(b) Zoom-in on cluster
Figure 66: Final plots of Osipkov-Merritt structure after G-perturbations. Again we see how the structure has expanded significantly.
(a) Initially the central potential is close to the value of -1
(b) Finally the central potential is seen in the semi-logarithmic plot to have a value of about -0.25.
Figure 67: Gravitational potential for Osipkov-Merritt data. First subplot is Φ shown versus radius, whereas the second plot is against logarithmic radius, to resolve the central region in greater detail. The lessening of the central potential is to be expected since we already saw in a previous plot how the structure expands and therefore feels a numerically smaller central gravitational potential.
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(a) Initially all velocities are almost identical
(b) The final picture is quite different. Now the velocities are completely different along each axis.
Figure 68: Velocities of particles , inside cluster of Osipkov-Merritt data, with respect to the clusters own velocity. From left to right we see particle velocities along the x, y and z-axis, all plotted as a function of the x-axis.
D.2
Line-of-sight
D.3
Removing gravitationally unbound particles from structures
To see the equilibrated structures from the final products more clearly, it is beneficial to remove the unbound particles which have escaped the gravitational potential of all the other particles. The total energy being positive is the condition which has to be met, in order to identify a free particle. This leads to the inequality Etot > 0 → Φ+ 12 v 2 > 0, where Φ is the gravitational potential of all the other q particles and v is the speed of the particle under investigation. It is computed as v = vx2 + vy2 + vz2 . In practice this has been solved by writing all the bound particles (where Φ + 21 v 2
